Question

Solve the following equation graphically & find area of region between lines &x axis 2x+y=6 & 2x-y=0​

Answer 1

We have, 2x + y = 6 ⇒ ⇒ y = 6 - 2x When x = 0, we have y = 6 - 2 x 0 = 6 When x = 3, we have y = 6 - 2 x 3 = 0 When x = 2, we have y = 6 - 2 x 2 = 2 Thus, we get the following table: x 0 3 2 y 6 0 2 Now, we plot the points A(0,6), B(3,0) and C(2,2) on the graph paper. We join A, B and C and extend it on both sides to obtain the graph of the equation 2x + y = 6. We have, 2x - y + 2 = 0 ⇒ ⇒ y = 2x + 2 When x = 0, we have y = 2 x 0 + 2 = 2 When x = -1, we have y = 2 x (-1) + 2 = 0 When x = 1, we have y = 2 x 1 + 2 = 4 Thus, we have the following table: x 0 -1 1 y 2 0 4 Now, we plot the points D(0,2), E(-1,0) and F(1,4) on the same graph paper. We join D,E and F and extend it on the both sides to obtain the graph of the equation 2x - y + 2 = 0. It is evident from the graph that the two lines intersect at point F(1,4). The area enclosed by the given lines and x-axis is shown in Fig. above Thus, x = 1, y = 4 is the solution of the given system of equations. Draw FM perpendicular from F on x-axis. Clearly, we have FM = y-coordinate of point F(1,4) = 4 and BE = 4

∴ Area of the shaded region = Area of △ △FBE ⇒

⇒ Area of the shaded region = 1 2 12(Base x Height) = 1 2 12(BE x FM) = ( 1 2 × 4 × 4 ) (12×4×4)sq. units = 8 sq.